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I've been working with hyperbolic functions and am completely confused by the Wikipedia definitions of hyperbolic sectors and angles. Are they the same thing? Based on my trial calculations, they seem to be quite different. So what are they, are they related, and how is the hyperbolic angle defined?

I'm assuming that the hyperbolic sector is the area enclosed by the x-axis, a ray meeting the hyperbola $x^2-y^2=1$ and the hyperbola.

More information edit: I'm reading from this link: http://www.mathed.soe.vt.edu/Undergraduates/EulersIdentity/HyperbolicTrig.pdf The formulas for hyperbolic trig functions can also be derived from the graph of a hyperbola. It says very specifically: "Let $alpha$ equal the hyperbolic angle between the x-axis and the point (x, y) on the unit regular hyperbola $x^2-y^2=1$. Then $alpha = 2A$ where $A$ is the area of the hyperbolic sector."

But in my experience, this is not true, leading once again to my question above.

Rohil Verma
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    "But in my experience, this is not true ..." What has your experience shown you? Note that hyperbolic angle measure is in hyperbolic radians with values defined as twice the area of the corresponding hyperbolic sector. Similarly (circular) radians can be defined as twice the area of the corresponding circular arc. Observations made in this answer to a related question may be helpful. – Blue Aug 05 '15 at 09:58

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The hyperbolic angle is the area of the hyperbolic sector taken twice:

https://upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Hyperbolic_functions-2.svg/275px-Hyperbolic_functions-2.svg.png

The hyperbolic argument is equal to the hyperbolic angle between the given vector and $X$ axis.

Thus this is the same as with complex numbers: the complex argument of $i$ is $\pi/2$, and the sector of $1/4$ of full circle has the area of $\pi/4$. Thus, the complex argument is also the area of the sector taken twice.

Anixx
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